Principal Bundle Structure of Matrix Manifolds - Marie Billaud-Friess (Mathematics, 2021)
In this paper, we introduce a new geometric description of the manifolds of matrices of fixed rank. The starting point is a geometric description of the Grassmann manifold <semantics>Gr(Rk)</semantics> of linear subspaces of dimension <semantics>r<k</semantics> in <semantics>Rk</semantics>, which avoids the use of equivalence classes. The set <semantics>Gr(Rk)</semantics> is equipped with an atlas, which provides it with the structure of an analytic manifold modeled on <semantics>R(k−r)×r</semantics>. Then, we define an atlas for the set <semantics>Mr(Rk×r)</semantics> of full rank matrices and prove that the resulting manifold is an analytic principal bundle with base <semantics>Gr(Rk)</semantics> and typical fibre <semantics>GLr</semantics>, the general linear group of invertible matrices in <semantics>Rk×k</semantics>. Finally, we define an atlas for the set <semantics>Mr(Rn×m)</semantics> of non-full rank matrices and prove that the resulting manifold is an analytic principal bundle with base <semantics>Gr(Rn)×Gr(Rm)</semantics> and typical fibre <semantics>GLr</semantics>. The atlas of <semantics>Mr(Rn×m)</semantics> is indexed on the manifold itself, which allows a natural definition of a neighbourhood for a given matrix, this neighbourhood being proved to possess the structure of a Lie group. Moreover, the set <semantics>Mr(Rn×m)</semantics> equipped with the topology induced by the atlas is proven to be an embedded submanifold of the matrix space <semantics>Rn×m</semantics> equipped with the subspace topology. The proposed geometric description then results in a description of the matrix space <semantics>Rn×m</semantics>, seen as the union of manifolds <semantics>Mr(Rn×m)</semantics>, as an analytic manifold equipped with a topology for which the matrix rank is a continuous map.